Precise control of lower limb exoskeletons during sit-to-stand (STS) transitions remains a central challenge in rehabilitation robotics owing to the highly nonlinear, time-varying dynamics of the human-exoskeleton system and the stringent trajectory tracking requirements imposed by clinical safety. This paper presents the systematic design, simulation, and comparative evaluation of three control strategies: a classical Proportional-Integral-Derivative (PID) controller, a Linear Quadratic Regulator (LQR), and a novel Hybrid PID-LQR controller applied to a bilateral lower limb exoskeleton performing the sit-to-stand transition. A high-fidelity, physics-based dynamic model of the exoskeleton is constructed by importing a SolidWorks CAD assembly directly into the MATLAB/Simulink Simscape Multibody environment, preserving accurate geometric and inertial properties of all links. Physiologically representative reference joint trajectories for the hip, knee, and ankle joints are generated using OpenSim musculoskeletal simulation and decomposed into three biomechanical phases: flexion-momentum (0-33%), momentum-transfer (34-66%), and extension (67-100%). The proposed Hybrid PID-LQR controller combines the optimal transient response of LQR with the integral disturbance rejection of PID through a tuned blending coefficient alpha = 0.65. Simulation results demonstrate that the Hybrid PID-LQR achieves RMSE reductions of 72.3% and 70.4% over PID at the hip and knee joints, respectively, reduces settling time by over 90% relative to PID across all joints, and limits overshoot to 2.39%-6.10%, confirming its superiority over both baseline strategies across all evaluated performance metrics and demonstrating strong translational potential for clinical assistive exoskeleton deployment.